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Arithmetic and Geometric Sequences and their Summation

Arithmetic and Geometric Sequences and their Summation. 14.1 Sequences. arithmetic sequence. geometric sequence. geometric sequence. geometric sequence. Find the next two terms of the following sequences : 2, 5, 8, 11,…… 2, 6, 18, 54, …. 2, 4, 8, 16,……. 5, -25, 125, -625, ….

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Arithmetic and Geometric Sequences and their Summation

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  1. Arithmetic and Geometric Sequences and their Summation

  2. 14.1 Sequences arithmetic sequence geometric sequence geometric sequence geometric sequence Find the next two terms of the following sequences : 2, 5, 8, 11,…… 2, 6, 18, 54, …. 2, 4, 8, 16,……. 5, -25, 125, -625, …. 3, 4, 6, 9, 13, ……. 5, 2, -1, -4, ….. 0, sin20o, 2sin30o, 3sin40o arithmetic sequence

  3. 14.1 Sequences Consider the following sequence:1, 3, 5, 7, 9, ….., 111 3 is the second term of the sequence, mathematically, T(2) = 3 or T2 = 3 1 is the first term of the sequence,mathematically, T(1) = 1 or T1 = 1 5 is the third term of the sequence, mathematically, T(3) = 5 or T3 = 5 111 is the nth term of the sequence, mathematically, T(n) = 111 or Tn = 111

  4. 14.1 Sequences Consider the sequence 2, 4, 8, 16, …. So, the sequence can be represented by the general term T(n) = 2n or Tn = 2n The sequence is formed from timing 2 to the previous term.

  5. P.159Ex. 14A

  6. 14.2 Arithmetic Sequence An arithmetic sequence(A.S. /A.P.) is a sequence having a common difference.

  7. 14.2 Arithmetic Sequence Illustrative Examples

  8. 14.2 Arithmetic Sequence

  9. 14.2 Arithmetic Sequence

  10. 14.2 Arithmetic Sequence

  11. P.166Ex. 14B

  12. 14.2 Arithmetic Sequence Arithmetic Means When a, b and c are three consecutive terms of arithmetic sequence, the middle term b is called the arithmetic mean of a and c.

  13. 14.2 Arithmetic Sequence Arithmetic Means Insert two arithmetic means between 11 and 35.

  14. 14.2 Arithmetic Sequence Insert two arithmetic means between 11 and 35.

  15. P.170Ex. 14C

  16. 14.3 Geometric Sequence A geometric sequence(G.S. / G.P.) is a sequence having a common ratio.

  17. 14.3 Geometric Sequence Illustrative Examples

  18. 14.3 Geometric Sequence

  19. 14.3 Geometric Sequence

  20. 14.3 Geometric Sequence

  21. P.176Ex. 14D

  22. 14.3 Geometric Sequence Geometric Means When x, y and z are three consecutive terms of geometric sequence, the middle term y is called the geometric mean of x and z.

  23. 14.3 Geometric Sequence Geometric Means Insert two geometric means between 16 and -54.

  24. 14.3 Geometric Sequence Insert two geometric means between 16 and -54.

  25. P.181Ex. 14E

  26. 14.4 Series The expression T(1) + T(2) + T(3) +….+ T(n) is called a series. We usually denote the sum of the first n term of a series by the notation S(n). Let’s consider a sequence : T(1), T(2), T(3), T(4), …., T(n)

  27. 14.5 Arithmetic Series Arithmetic Sequence : 2, 5, 8, 11, … Arithmetic Series : 2 + 5 + 8 + 11 + ….

  28. 14.5 Arithmetic Series Formula of Arithmetic Series S(n) = a + a + d + a + 2d + a + 3d + …. + a + (n - 1)d l

  29. 14.5 Arithmetic Series Formula of Arithmetic Series S(n) = l + l - d + l - 2d + l - 3d + …. + a + d+ a

  30. 14.5 Arithmetic Series S(n) = a + a + d + a + 2d + a + 3d + ………... + a + (n - 1)d S(n) = l + l – d + l - 2d + l - 3d + ….+ a + d+ a 2S(n) =(a + l)+(a + l)+(a + l)+(a + l)+….. +(a + l) 2S(n) = n(a + l)

  31. 14.5 Arithmetic Series

  32. P.189Ex. 14F

  33. 14.6 Geometric Series Geometric Sequence : 3, 9, 27, 81, … Geometric Series : 3 + 9 + 27 + 81

  34. 14.6 Geometric Series Formula of Geometric Series S(n) = a + aR + aR2 +aR3+ …. + aRn-1

  35. 14.6 Geometric Series Formula of Geometric Series R.S(n) = aR + aR2 + aR3 +aR4+ …. + aRn

  36. Subtracting two series S(n) = a + aR + aR2 +aR3+ …. + aRn-1 S(n) –R.S(n) = a - aRn R.S(n) = aR + aR2 + aR3 +aR4+ …. + aRn (1 – R) S(n) = a (1 – Rn)

  37. 14.6 Geometric Series Timing –1 on both numerator and denominator

  38. P.196Ex. 14G

  39. 14.6 Geometric Series Sum to Infinity of a Geometric Series

  40. 14.6 Geometric Series Sum to Infinity of a Geometric Series Consider such a Geometric Series What is the value of common ratio R ?

  41. 14.6 Geometric Series Sum to Infinity of a Geometric Series Consider Rn where n tends to the infinity

  42. What will occur for if n tends to infinity ? where –1< R <1

  43. Summation of a geometric Series to infinity

  44. P.203Ex. 14H

  45. (extension module) Summation Notation

  46. Consider the symbol where T( r ) = 3r + 5 = 3(1) + 5 + 3(2) + 5+3(3) + 5 + 3(4) +5 = 50

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